Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 739602, 17 pages
doi:10.1155/2008/739602
Research Article
On the Asymptotic Integration of
Nonlinear Dynamic Equations
Elvan Akın-Bohner,1 Martin Bohner,1 Smaı̈l Djebali,2 and Toufik Moussaoui2
1
Department of Mathematics and Statistics, Missouri University of Science and Technology,
Rolla, MO 65409-0020, USA
2
Départment de Mathématiques, École Normale Supérieure, P.O. Box 92, 16050 Kouba, Algiers, Algeria
Correspondence should be addressed to Martin Bohner, bohner@mst.edu
Received 25 June 2007; Revised 12 November 2007; Accepted 29 January 2008
Recommended by Ondřej Došlý
The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class
of second-order nonlinear dynamic equations on unbounded time scales. Four different results are
obtained by using the Banach fixed point theorem, the Boyd and Wong fixed point theorem, the
Leray-Schauder nonlinear alternative, and the Schauder fixed point theorem. For each theorem, an
illustrative example is presented. The results provide unification and some extensions in the time
scale setup of the theory of asymptotic integration of nonlinear equations both in the continuous
and discrete cases.
Copyright q 2008 Elvan Akın-Bohner et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
This work is devoted to the study of the existence and asymptotic behavior of solutions to the
nonlinear dynamic equation
uΔΔ ft, u 0,
t ∈ T,
1.1
where the function f : T × R → R is continuous and T is a time scale i.e., a nonempty closed
subset of the real numbers; see 1, 2 and Section 2 below that has a minimal element t0 > 0
and is unbounded above, that is,
lim tn ∞ for some set tn : n ∈ N ⊂ T.
1.2
n→∞
In this paper, we offer conditions that ensure that for given a, b ∈ R, there exists a solution u of
1.1 satisfying the asymptotic behavior
ut at b o1
as t −→ ∞.
1.3
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Advances in Difference Equations
In 3, the general solution of the linear dynamic equation
uΔΔ huσ 0,
t ≥ t0 ,
1.4
where uσ u ◦ σ, is proved to have the asymptotic representation 1.3 whenever
∞
σs|hs|Δs < ∞.
1.5
t0
The study is extended in 4 to the investigation of oscillatory solutions for the more general
dynamic equation
σ
uΔΔ htuΔ gt f ◦ uσ 0,
1.6
where the coefficients h and g satisfy some integral conditions. The existence of solutions converging to zero is considered in 5 for a linear nonhomogeneous dynamic equation in a selfadjoint form. In 6, the authors considered the nonlinear dynamic equation 1.1 for the time
scales T R and T k Z. They assumed the existence of some positive rd-continuous function
h and a positive nondecreasing function g with g0 0 and gu > 0 for u > 0 such that
|ft, u| ≤ htg
|u|
t
1.7
with
∞
htΔt < ∞.
1.8
Then, they obtained the linear behavior of solutions see 6, Theorem 4.1
lim
t→∞
ut
a,
t
a/
0.
1.9
We mention that the dynamic equation 1.1 contains as special cases both differential
T R and difference T Z equations of the form
u ft, u 0,
t ∈ R,
Δ2 u ft, u 0,
t ∈ Z.
1.10
7, Chapter 8 is entirely devoted to the asymptotic behavior of linear difference equations and
contains some classical and fundamental results. The mth order nonlinear difference equation
Δm u αn fu 0,
n ∈ N,
1.11
where m ∈ N, α : N → R, and f : R → R, is studied in 8. Sufficient conditions which guarantee
existence of solutions converging to some limit or having certain types of asymptotic behavior
are given. In the particular case of second-order difference equations m 2, a solution un is
shown to have the asymptotic representation see 8, Theorem 2, page 4692
un an b o1,
n ≥ n0 ,
1.12
Elvan Akın-Bohner et al.
3
provided that
n|αn | < ∞
1.13
n≥1
and both boundedness and uniform continuity of f are assumed. For further related results in
the discrete case, we refer the reader to 7–15. In the continuous case, that is, T R, the study
of the linear differential equation
u htu 0,
1.14
where
∞
t|ht|dt < ∞,
1.15
goes back to at least the works of Bôcher 16 and Dini 17 published at the beginning
of the twentieth century, and it was also adapted by Bellman 18 in 1947, where the limit
limt→∞ u t a is obtained yielding by L’Hospital’s rule the limiting behavior limt→∞ ut/t a. The nonlinear differential equation
u htgu 0
1.16
has been initially studied by Bihari in 1957 under 1.15 with further growth assumptions upon
the nonlinearity g see 19. The problem of the existence and extendability of solutions for
nonlinear ordinary differential equations has been widely investigated during the last couple
of years see, e.g., 20–22. Regarding the general theory of asymptotic integration of ODEs,
more details and recent developments may be found in the works 23–30 and the references
therein. Note also that 1.3 is referred to as Property L for the continuous case in 29, and it
seems that this notion was introduced first in 31.
Inspired and motivated by the results obtained both for difference and differential equations, our aim in this paper is to extend some of these results to nonlinear dynamic equations
on time scales. In order to obtain existence of global solutions and their asymptotic behavior at
positive infinity, we consider an arbitrary time scale unbounded above and we will be interested in the asymptotic behavior 1.3 of a solution u of 1.1. Here a and b are real numbers.
Considered in the spirit of the linear asymptotic conditions 1.9 and 1.12, the asymptotic development 1.3 will be used throughout this work. Indeed, 1.1 may be seen as a perturbation
of the homogeneous equation uΔΔ 0, the solutions of which are the straight lines ut at b.
Taking into account the restrictions 1.5, 1.8, 1.9, 1.12, 1.13, and 1.15, our results will
also depend heavily on the growth of the nonlinear function f with respect to the unknown u.
The setup of this paper is as follows. Section 2 contains some preliminary definitions and
results from the theory of time scales. In Section 3, we only state the main theorems. These are
four distinct results, each of which guarantees the existence of asymptotically linear solutions
according to 1.3. For the first two results, Lipschitz-like hypotheses are assumed on the nonlinearity, the third one is concerned with the sublinear growth case, while the fourth and last
one generalizes a result from 6. In the first three theorems, existence of solutions asymptotic
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Advances in Difference Equations
to any prescribed line is proved while in the last one, we describe linear behavior of some
solution. Section 4 features some examples that illustrate the applicability of the main results.
The proofs of the main results are presented in Section 5. They are based on the fixed point
theorems of Banach, Boyd and Wong, the Leray-Schauder nonlinear alternative, and Schauder,
respectively. We end this paper with some concluding remarks in Section 6.
2. Preliminaries
In this section, we gather some standard definitions, properties, and notations from the time
scales calculus see 1, 2.
Definition 2.1. Define the forward and backward jump operators σ : T → T and ρ : T → T by
σt : inf s > t : s ∈ T ,
ρt : sup s < t : s ∈ T ,
2.1
respectively. A function g : T → R is called rd-continuous if it is continuous at points t ∈ T with
σt t and if it has finite left-sided limits at points t ∈ T with ρt t.
Definition 2.2. For t ∈ T and a function g : T → R, define the delta derivative g Δ t to be the
number if it exists with the property that given ε > 0, there is a neighborhood U of t such that
|gσt − gs − g Δ tσt − s| < ε|σt − s|,
∀s ∈ U.
2.2
Define also the second delta derivative by g ΔΔ g Δ Δ .
Definition 2.3. If G is an antiderivative of g : T → R, that is, GΔ g, then the integral of g is
defined by
b
gtΔt Gb − Ga.z
2.3
a
Moreover, improper integrals are defined by
∞
T
gtΔt lim gtΔt.
a
2.4
T →∞ a
Remark 2.4. A well-known existence theorem 1, Theorem 1.74 says that rd-continuous functions possess antiderivatives.
Remark 2.5. Note that in the case T R, we have
σt ρt t,
f ΔΔ t f t,
f Δ t f t,
b
b
ftΔt ftdt,
a
2.5
a
and in the case T Z, we have
f Δ t Δft : ft 1 − ft,
b
b−1
f ΔΔ t ft 2 − 2ft 1 ft,
ftΔt ft if a < b.
σt t 1,
ρt t − 1,
a
ta
2.6
Elvan Akın-Bohner et al.
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Remark 2.6. In the theory of orthogonal polynomials and quantum calculus, an appropriate
time scale is T qZ {qk : k ∈ Z} ∪ {0}, where q > 1, and thus we have
σt qt,
f
ΔΔ
t ρt q − 1t,
f q2 t − 2fqt ft
q − 12 t2
t
,
f Δ t fqt − ft
,
q − 1t
2.7
logq t−1
fsΔs q − 1
1
q fq if t > 1,
i
i
i1
see 32, Lemma 2ii. In this case, 1.1 is called a q-difference equation.
We conclude this section with an auxiliary result that will be needed frequently for the
proofs of the main theorems in Section 5.
Lemma 2.7. Let g : T → 0, ∞ be rd-continuous and assume
∞
∗
G :
σs − t0 gsΔs < ∞.
2.8
t0
Then,
i
ii
iii
∞
t
σs − tgsΔs ≤ G∗ , for all t ≥ t0 ,
t
σs − tgsΔs → 0 as t → ∞,
t0
gsΔs < ∞.
∞
∞
Proof. For fixed T ∈ T and t ∈ t0 , T , 1, Theorem 1.117ii can be used to show that
T
Gt :
σs − tgsΔs implies GΔ t −
t
T
gsΔs ≤ 0
2.9
t
so that G is nonincreasing and hence Gt ≤ Gt0 ≤ G∗ . Now, let T → ∞ to obtain i. Next,
0≤
∞
σs − tgsΔs ≤
∞
t
σs − t0 gsΔs −→ 0 as t −→ ∞
2.10
t
so that ii holds. Finally, for sufficiently large T ∈ T, let t∗ ∈ t0 1, T ∩ T so that
T
gsΔs t0
≤
T
t∗
T
gsΔs gsΔs
t0
σs − t0 gsΔs t∗
≤ G∗ t∗
gsΔs
t0
t∗
gsΔs.
t0
Letting T → ∞, we derive iii.
t∗
2.11
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Advances in Difference Equations
3. Main results
Throughout this paper, for a given nonnegative rd-continuous function h : T → R, we consider
when they exist the constants
∞
∞
∗
∗∗
H :
σs − t0 hsΔs,
H :
σs − t0 shsΔs.
3.1
t0
t0
We are now in position to state the four main results of this paper.
Theorem 3.1. Assume
∃L > 0 with
∞
σs − t0 |fs, 0|Δs ≤ L,
3.2
t0
|ft, u1 − ft, u2 | ≤ ht|u1 − u2 |,
H ∗ < 1,
∀t ≥ t0 , u1 , u2 ∈ R,
H ∗∗ < ∞.
3.3
3.4
Then, for all a, b ∈ R, 1.1 has a solution u on t0 , ∞ satisfying 1.3.
Theorem 3.2. Assume H ∗ < ∞, 3.2, and
ft, u1 − ft, u2 ≤ ht
u1 − u2 ,
H ∗ u1 − u2 ∀t ≥ t0 , u1 , u2 ∈ R.
3.5
Then, the conclusion of Theorem 3.1 holds true.
It is clear that 3.5 is stronger than 3.3 of Theorem 3.1. However, the assumption H ∗ <
∞ in Theorem 3.2 is weaker than the restriction H ∗ < 1 in 3.4 and no further restriction is
made on the second integral H ∗∗ .
Theorem 3.3. Assume 3.4 and
|ft, u| ≤ ht|u|,
∀t, u ∈ T × R.
3.6
Then, the conclusion of Theorem 3.1 holds true.
In the last existence result, we are rather concerned with existence of at least one solution
asymptotic to a specified line.
Theorem 3.4. Assume H ∗ < ∞ and suppose there exists a nondecreasing function g : 0, ∞ → 0, ∞
such that
|u|
, ∀t, u ∈ T × R.
|ft, u| ≤ htg
3.7
t
Suppose also that there exist a, b ∈ R, K > 0, and t∗ ≥ t0 such that
K |b|
H ∗ sup gs : 0 ≤ s ≤ ∗ ∗ |a| ≤ K.
t
t
Then, 1.1 has a solution u on t∗ , ∞ satisfying 1.3.
3.8
Elvan Akın-Bohner et al.
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4. Examples
In this section, we illustrate each of the four theorems given in Section 3 by means of an example.
Example 4.1 application of Theorem 3.1. Let γ : R → R satisfy
γ u1 − γ u2 ≤ u1 − u2 , ∀u1 , u2 ∈ R,
γ0 1,
4.1
for example, γu arctanu 1. By Theorem 3.1, for any a, b ∈ R, the difference equation
Δ2 u γu
0,
4t2 2t
t ∈ T N,
4.2
has at least one solution u on N satisfying 1.3. Indeed, let
ht 1
,
4t2 2t
ft, u htγu,
1
L .
4
According to 4.1, clearly 3.3 is satisfied as well as 3.2:
∞
∞
∞ n
1
1 1 n 1
1
1
σs − 1|fs, 0|Δs ≤
L.
4 n1 n 2
4 n1 2
4
1
4.3
4.4
Moreover,
∗
0<H ≤H
∗∗
∞
1
n
∞
1
1
2 1
σs − 1shsΔs n
<1
2
2
4
4n
n1
4.5
so that 3.4 also holds.
Example 4.2 application of Theorem 3.2. Let T be any time scale which is unbounded above
such that its graininess is bounded above. Suppose also 1 ∈ T. Let p > 1. By 2, Example 5.72,
∞
Δs
M :
< ∞.
4.6
p
1 s
Let γ : R → R be such that
γ0 0,
γ u1 − γ u2 ≤
u1 − u2 ,
M u1 − u2 ∀u1 , u2 ∈ R,
4.7
for example, γu |u|/M |u|:
Mu1 − u2 |γu1 − γu2 | M u1 M u2 Mu1 − u2 ≤
M2 M u1 u2 u1 − u2 ≤
M u1 − u2 γ u1 − u2 .
4.8
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Advances in Difference Equations
By Theorem 3.2, for any a, b ∈ R, the dynamic equation
uΔΔ γu
0,
tp σt
4.9
t ≥ 1, t ∈ T,
has at least one solution u on T satisfying 1.3. To prove this, let
ht 1
tp σt
,
ft, u htγu,
L 1.
4.10
Δs
M < ∞.
sp
4.11
Now note that
H∗ ∞
σs − 1hsΔs ≤
1
∞
σshsΔs ∞
1
1
Thus, together with 4.7, we clearly have 3.5 and 3.2.
Example 4.3 application of Theorem 3.3. For any a, b ∈ R, the dynamic equation
uΔΔ ht ln1 |u| 0,
t ≥ t0 , t ∈ T,
4.12
has a solution satisfying the asymptotic representation 1.3 provided 3.4 is fulfilled, for example, ht 1/4t2 2t . This follows directly from Theorem 3.3.
Example 4.4 application of Theorem 3.4. Let q > 1. By Theorem 3.4, the q-difference equation
uΔΔ e|u|/t
t logq t2
t ∈ qN ,
0,
4.13
has at least one solution u on qN which behaves as ut t when t → ∞. In fact, setting
ht 1
,
t logq t2
gs es ,
ft, u htg
|u|
,
t
4.14
we can first see refer to Remark 2.6 that the integral
∗
H ∞
σs − qhsΔs ≤ q
q
∞
q
∞
1
Δs
qq
−
1
2
2
n
slogq s
n1
Δs
4.15
converges. Moreover, 3.7 is clearly satisfied. Let b 0 and K H ∗ e2|a| . Since T is unbounded
above, there exists α ≥ 1 such that t∗ αK ∈ T. Then,
K |b|
1
∗
s
H sup gs : 0 ≤ s ≤ ∗ ∗ |a| H sup e : 0 ≤ s ≤ 1 |a| t
t
α
∗
∗ 1|a|1/α
H e
so that 3.8 holds true as well.
∗ 2|a|
≤H e
K
4.16
Elvan Akın-Bohner et al.
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5. Proofs
For any a, b ∈ R, consider the transformation vt ut − at − b. Then, u is a solution of 1.1
if and only if v is a solution of
vΔΔ t ft, vt at b 0,
t ≥ t0 .
5.1
Consider the space
C0 : C0 t0 , ∞ T v ∈ C t0 , ∞ T , R : lim vt 0 ,
t→∞
5.2
where t0 , ∞T : t0 , ∞ ∩ T. Endowed with the supnorm v sup{|vt| : t ≥ t0 }, C0 is a
Banach space. Define the mapping A for v ∈ C0 if the improper integral exists by
Avt ∞
t − σs f s, vs as b Δs.
5.3
t
It is clear that fixed points of the operator A are solutions of 5.1. Observe also that Av is
continuous if v is continuous, since then note that f is assumed to be continuous and that
σ is rd-continuous an rd-continuous function is integrated note that this is possible due to
Remark 2.4 which yields a delta differentiable and hence a continuous function; see 1, Theorem 1.16i. Now, we are ready to prove the four results giving not only asymptotic behavior
but also existence of global solutions.
5.1. Result based on the Banach fixed point theorem
The proof of Theorem 3.1 relies on the Banach fixed point theorem, which we recall here for
completeness.
Theorem 5.1 the Banach fixed point theorem. Let X be a Banach space and let A : X → X be a
contraction. Then, A has a unique fixed point in X.
Proof of Theorem 3.1. Let v ∈ C0 . First, we use 3.3 to find
f s, vs as b − fs, 0 ≤ hsvs as b ≤ hs v |a|s |b|
5.4
so that
Avt ≤
∞
σs − tfs, vs as b − fs, 0Δs t
≤ v |b|
∞
σs − tfs, 0Δs
t
∞
t
∞
∞
σs − t hsΔs |a|
σs − t shsΔs σs − t fs, 0Δs,
t
t
5.5
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Advances in Difference Equations
which tends to 0 as t → ∞ when applying 3.2 and 3.4 together with Lemma 2.7ii three
times. Hence, Av ∈ C0 and therefore A : C0 → C0 . Moreover, passing to the supremum above
and using Lemma 2.7i three times, we also find that A is indeed well defined and that
Av ≤ v |b| H ∗ |a|H ∗∗ L.
5.6
Next, let v1 , v2 ∈ C0 . With 3.3, we get
f s, v1 s as b − f s, v2 s as b ≤ hsv1 − v2 5.7
so that
|Av1 t − Av2 t| ≤ v1 − v2 ∞
σs − t hsΔs ≤ H ∗ v1 − v2 ,
5.8
t
where we used the first part of 3.4 together with Lemma 2.7i. Passing to the supremum, we
get
Av1 − Av2 ≤ H ∗ v1 − v2 ,
5.9
and due to the first part of 3.4, A is a contraction. According to Theorem 5.1, A has a fixed
point in C0 .
5.2. Result based on the Boyd and Wong fixed point theorem
To prove Theorem 3.2, we employ the Boyd and Wong fixed point theorem from 33, which
extends Theorem 5.1 and is recalled here together with a pertinent definition for completeness.
Definition 5.2. Let X be a Banach space and let A : X → X be a mapping. A is said to be a
nonlinear contraction if there exists a continuous nondecreasing function ψ : 0, ∞ → 0, ∞
such that ψ0 0 and ψx < x for all x > 0 with the property
Au − Av ≤ ψ u − v ,
∀u, v ∈ X.
5.10
Theorem 5.3 the Boyd and Wong fixed point theorem. Let X be a Banach space and let A : X →
X be a nonlinear contraction. Then, A has a unique fixed point in X.
Proof of Theorem 3.2. Let v ∈ C0 . From 3.5, we infer the estimates
|f s, vs as b − fs, 0 ≤ hs
|vs as b|
≤ hs
|vs as b|
H∗
5.11
so that
|Avt| ≤
∞
σs − t f s, vs as b − fs, 0Δs σs − t hsΔs t
≤
∞
t
∞
t
∞
t
σs − t fs, 0Δs,
σs − t fs, 0Δs
5.12
Elvan Akın-Bohner et al.
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which tends to 0 as t → ∞ when applying 3.2 and H ∗ < ∞ together with Lemma 2.7ii twice.
Hence, Av ∈ C0 and therefore A : C0 → C0 . Furthermore, using Lemma 2.7i twice, we find
that A is well defined and that
Av ≤ H ∗ L.
5.13
We introduce a continuous nondecreasing function ψ : 0, ∞ → 0, ∞ satisfying ψ0 0 and
ψx < x, for all x > 0 by
ψx H ∗x
,
H∗ x
∀x ≥ 0.
5.14
Let v1 , v2 ∈ C0 . Assumption 3.5 yields that
f s, v1 s as b − f s, v2 s as b ≤ hs ψ v1 − v2 H∗
5.15
so that
Av1 t − Av2 t ≤ ψ v1 − v2 ∞
t
σs − t
hs
Δs ≤ ψ v1 − v2 ,
H∗
5.16
where we used 0 < H ∗ < ∞ together with Lemma 2.7i. Passing to the supremum, we get
Av1 − Av2 ≤ ψ v1 − v2 ,
5.17
and by Definition 5.2, A is a nonlinear contraction. According to Theorem 5.3, A has a fixed
point in C0 .
5.3. Result based on the Leray-Schauder nonlinear alternative
The celebrated Leray-Schauder nonlinear alternative see, e.g., 34 is fundamental in the
proof of Theorem 3.3. Recall that an operator is said to be completely continuous if it is continuous and maps bounded sets into relatively compact sets.
Theorem 5.4 the Leray-Schauder nonlinear alternative. Let X be a Banach space, Ω ⊂ X bounded
and open, 0 ∈ Ω, and A : Ω → X a completely continuous operator. Then, either there exist u ∈ ∂Ω
and λ > 1 such that Au λu or A has a fixed point in Ω.
We need the time scales version of the compactness criterion for subsets of C0 which is
due to Avramescu for the case T R see 20, 35.
Proposition 5.5. Assume that the subset B ⊂ C0 has the following properties:
i B is uniformly bounded, that is, there exists a constant N > 0 with
|ut| ≤ N,
∀t ≥ t0 , u ∈ B,
5.18
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Advances in Difference Equations
ii B is equicontinuous, that is, for every ε > 0 there exists δε > 0 with
ut1 − ut2 < ε,
∀t1 , t2 ≥ t0 ,
|t1 − t2 | < δε,
u ∈ B,
5.19
iii B is equiconvergent, that is, for every ε > 0 there exists t∗ ε > t0 with
|ut| < ε,
∀t ≥ t∗ ε, u ∈ B.
5.20
Then, B is relatively compact.
Proof. Following 36, proof of Proposition 2.2, consider an interval α, βT α, β ∩ T, α < β,
and C Cα, βT , R. The spaces C0 and C are isomorphic by the mapping Φ defined by
Φxt x ϕt , if t ∈ α, βT ,
x∞,
if t β,
5.21
where ϕ : α, βT → T is a continuous, strictly nondecreasing function with limt→β− ϕt ∞.
From ii and iii, B is equicontinuous in C0 . Then, ΦB is equicontinuous and uniformly
bounded in C. By the Arzelà-Ascoli theorem for time scales 37, Lemma 2.6, we conclude that
ΦB is relatively compact in C, which completes the proof.
Proof of Theorem 3.3. Define
β : |a|H
∗∗
∗
|b|H ,
β1 − H ∗ −1 , if β / 0,
m :
1,
if β 0.
5.22
We also introduce
Ω : v ∈ C0 : v < m
5.23
and note that Ω ⊂ C0 is open and by 3.4 satisfies 0 ∈ Ω since m > 0. Let v ∈ Ω. From 3.6, we
get
f s, vs as b ≤ hsvs as b ≤ hs m |b| |a|s
5.24
so that
Avt ≤ m |b|
∞
t
∞
σs − thsΔs |a|
σs − t shsΔs,
5.25
t
which tends to 0 as t → ∞ when applying 3.4 together with Lemma 2.7ii twice. This means
that AΩ is equiconvergent observe Proposition 5.5iii and that Av ∈ C0 and therefore
A : Ω → C0 . By Lemma 2.7i, A is well defined, and passing to the supremum in 5.25, we
get
Av ≤ m |b| H ∗ |a|H ∗∗ mH ∗ β.
5.26
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13
We conclude that AΩ is uniformly bounded observe Proposition 5.5i. Now use 5.24
again to deduce
AvΔ t ∞ f s, vs as b Δs ≤ m |b| ∞ hsΔs |a| ∞ shsΔs.
t
t0
t0
5.27
Using 3.4 and Lemma 2.7iii twice, we find that the right-hand side above is equal to a finite
constant, say R. Thus,
Av t2 − Av t1 ≤ Rt2 − t1 −→ 0 as t2 −→ t1
5.28
and so AΩ is equicontinuous observe Proposition 5.5ii. Altogether, by Proposition 5.5,
AΩ is relatively compact.
It remains to prove that A is continuous. Let v ∈ Ω and let vn ⊂ Ω be a sequence
converging strongly to the limit v, that is, vn − v → 0 as n → ∞. By 5.24, we have the
estimate
f s, vn s as b ≤ m |b| |a|s hs.
5.29
Since H ∗ < ∞ and H ∗∗ < ∞ and because of Lemma 2.7i, we infer from the Lebesgue dominated convergence theorem see 38 and 37, Theorem 10.1 that
∞
lim
n→∞
σs − tfs, vn s as bΔs t
∞
σs − tfs, vs as bΔs.
5.30
t
Then, Avn → Av pointwise as n → ∞. In addition, AΩ is relatively compact. Then, there
exists a subsequence Avnk of Avn converging strongly to a certain w ∈ C0 . As the strong
convergence implies the pointwise convergence keeping the limit function, we find that w Av. Now, Avn converges strongly to Av as n → ∞ and thus the mapping A is continuous.
Altogether, A : Ω → C0 is completely continuous. Let v ∈ ∂Ω and λ > 1 be such that
Av λv. Then, using 5.26,
λm λv Av ≤ H ∗ m β
5.31
so that
β
H ∗,
1<λ≤H 1,
m
∗
if β 0
if β /
0
≤ 1,
a contradiction. Hence, by Theorem 5.4, A has a fixed point in Ω.
5.32
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Advances in Difference Equations
5.4. Result based on the Schauder fixed point theorem
To prove Theorem 3.4, we appeal to the Schauder fixed point theorem see, e.g., 34.
Theorem 5.6 the Schauder fixed point theorem. Let X be a Banach space and let B ⊂ X be
nonempty, bounded, closed, and convex. Let A : B → B be a completely continuous operator. Then, A
has a fixed point in B.
Proof of Theorem 3.4. Consider the closed ball B : {v ∈ C0 : v ≤ K} and define
K |b|
θ : sup gu : 0 ≤ u ≤ ∗ ∗ |a| .
t
t
5.33
Let v ∈ B. By 3.7, we find
f s, vs as b ≤ hsg |vs as b| ≤ θhs,
s
s ≥ t∗ ,
since for s ≥ t∗ we have
vs as b v |b|
K |b|
K |b|
0≤
|a|.
≤
|a| ≤
|a| ≤
s
s
s
t∗
5.34
5.35
By 5.34, for t ≥ t∗ ,
Avt ≤
∞
σs − t f s, vs as b Δs ≤ θ
t
∞
σs − t hsΔs,
5.36
t
which tends to 0 as t → ∞ when applying H ∗ < ∞ together with Lemma 2.7ii. This means
that AB is equiconvergent observe Proposition 5.5iii and that Av ∈ C0 and therefore
A : B → C0 . Thanks to Lemma 2.7i, we also get that A is well defined and, passing to the
supremum above, we have
Av ≤ θH ∗ ≤ K,
5.37
where we used 3.8. We conclude that A : B → B and that AB is uniformly bounded
observe Proposition 5.5i. Next, let t1 , t2 ∈ T be such that t2 ≥ t1 ≥ t∗ . Then,
Avt2 − Avt1 ∞
∞
t2 − σs f s, vs as b Δs −
t1 − σs f s, vs as b Δs
t2
∞
t1
t2 − σs − t1 − σs f s, vs as b Δs −
t2
t1 − σs f s, vs as b Δs
t1
t2
∞
t2
t2 − t1 f s, vs as b Δs σs − t1 f s, vs as b Δs.
t2
t1
5.38
Elvan Akın-Bohner et al.
15
Using again 5.34, we find
Av t2 − Av t1 ≤ t2 − t1 θ
∞
hsΔs θ
t2
σshsΔs,
5.39
t1
t0
which tends to zero as t2 → t1 due to H ∗ < ∞ and Lemma 2.7iii for the second integral, use
that σh is rd-continuous and hence has an antiderivative Q by Remark 2.4, and thus this integral equals to Qt2 − Qt1 and Q is continuous. Therefore, AB is equicontinuous observe
Proposition 5.5ii. Altogether, by Proposition 5.5, AB is relatively compact. As in the proof
of Theorem 3.3, we may check that A is continuous. Thus, A : B → B is completely continuous.
According to Theorem 5.6, A has a fixed point in B.
6. Concluding remarks
In this work, specific results regarding the asymptotic behavior of the nonlinear dynamic
equation 1.1 have been obtained, extending some known results in the theories of difference and differential equations, for example to q-difference equations see Remark 2.6 and to
other cases of arbitrary time scales. Not only did our work extend the continuous and the discrete, but it also unified those two important cases and illuminated the common grounds of
the corresponding differential and difference equations. As a fundamental contribution to the
now well-established theory of time scales, it is hoped that our results will advance the area
and stimulate future research on this and related topics. For example, the more general case of
delta-derivative depending nonlinearity f ft, u, uΔ may be treated in an analogous manner
yielding the asymptotic behavior 1.3. For this purpose, additional restrictions on the growth
of f with respect to the derivative uΔ need to be assumed. The space C0 introduced in Section 5
is then extended to a space involving also the limit at infinity of the delta derivative; accordingly, a new compactness criterion is required. Also notice that the most informative condition
is 3.7 which shows how the nonlinearity grows in terms of the ratio u/t.
Apart from Theorem 3.4, Theorems 3.1, 3.2, and 3.3 are concerned with what is usually
called the inverse problem of seeking a solution asymptotic to a given line see 28, 29. We
point out that further to the asymptotic behavior, these theorems also provide existence of
solutions to initial value problems for the dynamic equation 1.1. Moreover, the existence of
solutions with behavior described by 1.3 does not mean that all solutions behave in the same
manner as shown in the nonlinear ordinary differential equation u 3/t5 u2 , t ≥ 1 see also
28, Section 5. Indeed, this equation admits by Theorem 3.1 a solution having Property L
while the solution ut 2t3 has not.
Finally, we mention that similar Bihari-type existence results of solutions which can be
expanded asymptotically as ut P t ot near positive infinity may also be obtained for
the nonhomogeneous dynamic equation
uΔΔ ft, u pt,
t ∈ T,
6.1
where P ΔΔ p and p is a polynomial. For the case T R, we refer to 24, Section 8, Theorem
18 see also 30.
16
Advances in Difference Equations
Acknowledgments
S. Djebali would like to thank his laboratory, Èquations aux Dèrivèes Partielles & Histoire des
Mathèmatiques, for supporting this work. M. Bohner acknowledges support by NSF Grant no.
0624127. The authors are grateful to two anonymous referees for their careful reading of the
first version of this manuscript and their constructive comments. Their suggestions substantially influenced the final presentation of the authors’ results.
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